The picture shows the back (i.e. rail) of a chair and the mechanism by which it was fastened to the stile. I'm not sure how those spiky pieces were initially inserted. Perhaps they "opened up" like a toggle bolt. In any case, I don't think that hammering them back in would make for a reliable connection. Given where I'm at, what course of action would you recommend for fixing the chair? My current plan is to used a dowel and then hope to find screw of the right depth and head.
Those do look like expansion anchors, AKA expansion bolts, as you state. However, because of the very shallow depth of the holes (which led to a lack of strength and early failure), it might be difficult to find a replacement that would hold.
If appearance is not important, I'd drill through the hole and use two nuts, with washers, to hold each screw. Longer screws might be needed, in that case, so there might be no need to match the threads of the existing screws.
If appearance is important, you could try forcing the cone nut out of the shell, and then squeezing the prongs of the shell closed with pliers or vise. Repair the damaged wood and reinforce, perhaps with epoxy or a metal plate on the hidden side, re-drill the holes, and reuse the old expansion bolt. How well it will hold depends on use and strength of repaired wood, and this would not be as strong as a through-bolt.